Optimal. Leaf size=58 \[ -\frac {3 b^3 \sin (c+d x) \, _2F_1\left (\frac {1}{2},\frac {13}{6};\frac {19}{6};\cos ^2(c+d x)\right )}{13 d \sqrt {\sin ^2(c+d x)} (b \sec (c+d x))^{13/3}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {16, 3772, 2643} \[ -\frac {3 b^3 \sin (c+d x) \, _2F_1\left (\frac {1}{2},\frac {13}{6};\frac {19}{6};\cos ^2(c+d x)\right )}{13 d \sqrt {\sin ^2(c+d x)} (b \sec (c+d x))^{13/3}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 16
Rule 2643
Rule 3772
Rubi steps
\begin {align*} \int \frac {\cos ^2(c+d x)}{(b \sec (c+d x))^{4/3}} \, dx &=b^2 \int \frac {1}{(b \sec (c+d x))^{10/3}} \, dx\\ &=\left (b^2 \left (\frac {\cos (c+d x)}{b}\right )^{2/3} (b \sec (c+d x))^{2/3}\right ) \int \left (\frac {\cos (c+d x)}{b}\right )^{10/3} \, dx\\ &=-\frac {3 \cos ^5(c+d x) \, _2F_1\left (\frac {1}{2},\frac {13}{6};\frac {19}{6};\cos ^2(c+d x)\right ) (b \sec (c+d x))^{2/3} \sin (c+d x)}{13 b^2 d \sqrt {\sin ^2(c+d x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.09, size = 60, normalized size = 1.03 \[ -\frac {3 b^2 \sqrt {-\tan ^2(c+d x)} \cot (c+d x) \, _2F_1\left (-\frac {5}{3},\frac {1}{2};-\frac {2}{3};\sec ^2(c+d x)\right )}{10 d (b \sec (c+d x))^{10/3}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.87, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\left (b \sec \left (d x + c\right )\right )^{\frac {2}{3}} \cos \left (d x + c\right )^{2}}{b^{2} \sec \left (d x + c\right )^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos \left (d x + c\right )^{2}}{\left (b \sec \left (d x + c\right )\right )^{\frac {4}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 1.03, size = 0, normalized size = 0.00 \[ \int \frac {\cos ^{2}\left (d x +c \right )}{\left (b \sec \left (d x +c \right )\right )^{\frac {4}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos \left (d x + c\right )^{2}}{\left (b \sec \left (d x + c\right )\right )^{\frac {4}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\cos \left (c+d\,x\right )}^2}{{\left (\frac {b}{\cos \left (c+d\,x\right )}\right )}^{4/3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos ^{2}{\left (c + d x \right )}}{\left (b \sec {\left (c + d x \right )}\right )^{\frac {4}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________